Questions tagged [dual-spaces]
The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.
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Show that $A^o$ is convex, balanced, closed in $A'$
Let $E$ is a normed space and $A \subset E$. Define
$$A^o := \{y \in A': |y(x)| \leq 1, \forall x \in A\}.$$
in which, $A'$ is the dual space of $A$.
With this definition, how to show that $A^o$ is ...
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0answers
19 views
Surjectivity of a certain map $\ell^1\to(\ell^\infty)'$ [duplicate]
I am studying dual spaces at the moment and I want to prove the following:
Proposition: The map $$\Phi_\infty:\ell^1\to(\ell^\infty)',\qquad(\Phi_\infty y)(x)=\sum_{i\in\mathbb{N}}x^iy^i$$ is not ...
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1answer
47 views
$B$ Banach $\implies B^*$ Banach.
Since we only need to check that $B^*$ is complete, we should prove Cauchy sequence $\lbrace l_n \rbrace$ converges.
In general idea, $$|(l-l_n)(f)| \leq |(l-l_m)(f)| + |(l_m-l_n)(f)| \leq |(l-l_m)(f)...
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0answers
14 views
surjectivity of dual operator congruence implies reflexivity
I tried to prove the following:
Let $X,Y$ be normed spaces and $\Phi:\mathcal{L}(X,Y)\to\mathcal{L}(Y^*,X^*)$, $\Phi(A)=A^*$ is surjective. Then $Y$ is reflexive. Here $X^*$ $(A^*)$ denotes the dual ...
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3answers
55 views
Proof that the following map $\Phi:\ell^1\to(\ell^\infty)'$ is not surjective
I am working on the dual spaces of sequence spaces, and I want to show that the map $$
\Phi:\ell^1\to(\ell^\infty)',\qquad(\Phi y)(x)=\sum_{i\in\mathbb{N}}y_ix_i
$$
is not surjective. I have already ...
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1answer
22 views
When are the topological duals $A^*$ and $B^*$ isomorphic?
If I have to normed vector spaces $A$ and $B$, I was wondering when the topological duals are isomorphic (i.e. $A^* \cong B^*)$ . Is it sufficient that $A \cong B$? Or that $A$ has to be isometric to $...
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1answer
26 views
Duality: dual maps and linear maps
Suppose we have $V$ and $W$ be $2$ $K$-vectorspaces and $f:V\rightarrow W$ is a linear map then:
For all $\varphi\in W^\ast$ one has $\varphi\circ f\in V^*$
The map $f^\ast:W^\ast\rightarrow V^\ast:\...
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1answer
31 views
Dual map and solving linear equation
Let $f:V\to W$ be a linear transformation from a vector space $V$ to a vector space $W$. Suppose that $b\in W$ satisfies $\phi(b)=0$ for all $\phi\in\ker(f^*)$. Show that there exists $x\in V$ such ...
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1answer
28 views
Given a linear operator $T$ and a linear functional $\phi_n(x)=(T(x))(n)$, show that $T$ is continuous iff $\phi \in X*$
Given $X$ a Banach space, and $T:X \rightarrow l_p$ a linear operator, with $1 \leq p \leq \infty$, for all $n \in \mathbb{N}$ consider the linear functional $\phi_n:X \rightarrow \mathbb{K}$, defined ...
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2answers
47 views
Dual space of $L^p(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$.
I want to show that for $p\in(1,+\infty)$ the dual space of $L^p(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$ is isometrically isomorphic to $L^q(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$, where $\frac{1}{p}+\...
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0answers
18 views
intersection in a simplex
In a triangulation $\Gamma$ of a (oriented) 2-manifold, consider a 2-simplex labeled by ($123$), where $1,2,3$ denote the order of vertices. Consider the dual $\Gamma^*$ of $\Gamma$, and then denote ...
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1answer
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$y \in \mathbb{K^N}$ a scalar sequence, if $\sum_{n \geq 1} x(n)y(n)$ is bounded,does $x \in l_{p^*}$? [on hold]
Given $y \in \mathbb{K^N}$ a scalar sequence, and $1<p<\infty$.
Suppose that, $\forall x \in l_p$, the series $\sum_{n \geq 1} x(n)y(n)$ is bounded. Show that $x \in l_{p^*}$.
With $l_{p^*}$ ...
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2answers
38 views
Show that a Banach space $X$ is not reflexive if a closed subspace of its dual separates the points of $X$
Given a Banach space $X$ and a closed subspace $Z$ of $X^*$ such that $Z \neq X^*$, suppose that $Z$ separates the points of $X$, I mean:
$x \in X, \, \, \, x^*(x) = 0 \, \, \forall x^* \in Z \...
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2answers
64 views
Which vector spaces are algebraic dual spaces?
Let us say that a vector space $V$ is an algebraic dual space if there exist a vector space $U$ such that $V$ is isomorphic to $U^*$, the vector space of all linear maps from $U$ to the corresponding ...
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1answer
45 views
The relationship between vector space dualization and matrix transposition.
Let $\mathbb{F}$ be a field. The category of matrices $\mathbf{Mat}$ has $\mathbb{N}_0$ as class of objects and $\mathrm{hom}(n,m)=\mathbb{F}^{n\times m}$ for $n,m\in\mathbb{N}_0$, with $\mathrm{id}_n=...
2
votes
2answers
42 views
Finding $B^*$, the dual basis
Find a basis $B$ for
$$V = \left\{ \left[
\begin{array}{cc}
x\\
y\\
z
\end{array}
\right] \in \mathbb{R}^3 \vert x+y+z = 0\right\}$$ and then find $B^*$, the dual basis for $B$.
The way we ...
3
votes
1answer
52 views
Dual of a subspace
In question Can a subspace have a larger dual?, they discussed the following type of situtation:
Suppose that $Y\subset X$ is a proper embedding of Banach spaces (i.e. the inclusion of a proper ...
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1answer
36 views
About weak convergence in $L^{\infty}$
doing my homework I'm dealing with this:
Let, for all $n\in \mathbb{N} \quad f_n(t) := e^{-nt^2}, \quad t \in [-1,1]$ Show that
1)$f_n \overset{\ast}{\rightharpoonup} 0$ in $L^\infty(-1,1)$
2)$f_n$ ...
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1answer
24 views
The dual basis spans the vector space of linear transformations.
Proof attempt:
Let $T \in L(V,\Bbb{F})$ where $L(V,\Bbb{F})$ is the space of linear functionals of $V$ and where $\Bbb{F}$ is some field. Let $\mathscr{F}=\{\Phi_1,\cdots \Phi_n\}$ be the dual basis ...
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0answers
38 views
On linear, continuous, injective map $T:V \to W$ with closed/ dense image
Let $V,W$ be normed linear spaces and $T:V \to W$ be a linear continuous, injective map .
Then is $T^{**}: V^{**} \to W^{**}$ is injective ? If this is not true in general, what if we also assume $Im ...
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2answers
34 views
Killing form for a different representation
In this Math Overflow question, the OP was asking if the Killing form defined for a different representation (than the adjoint) was related to the normal Killing form (defined w.r.t. the adjoint rep.):...
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0answers
29 views
Example of $U \subseteq V$ such that $V$ is infinite-dimensional and $U^0 = V'$ but $U \neq \{0\}$.
I am working through Axler's Linear Algebra Done Right, where he uses the notation $U^0$ for the annihilator subspace of the dual space $V'$ such that if $\varphi \in U^0$, then $U \subseteq \text{...
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votes
2answers
66 views
How to proof that the $(M^\perp)^\perp = \operatorname{span}(M)$? [closed]
I have some problems with the following problem. It seems to be so obvious that I don't just how to show that that is true. Could someone help me?
Let $M \subset \mathbb{R}^n$ be a nonempty subset....
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0answers
28 views
irreducible contragredient representation
I am trying to understand the following statement:
Suppose $\rho: G \longrightarrow \text{GL}_\mathbb{C}(V)$ an irreducible representation of a finite group $G$. Then its contragredient ...
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1answer
31 views
Proving that if the dual space $X^*$ of a Banach space $X$ is separable , then $X$ is separable
I have reading an answer post before , and there's something I do not understand .
For each $f_n$ in the unit ball of $X^*$ , why there exist $x_n \in X$ with $\|x_n \| \le1$ such that $f_n(x_n) \ge \...
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0answers
15 views
Question about smoothness of a Banach space.
Define $\delta: [0,2] \to [0,1]$ defined by
$$\delta_U(\epsilon) = \inf \bigg\{\frac{1}{2}\bigg(2-\|u_1+u_2\|\bigg): \ u_1, u_2 \in U^0, \|u_1-u_2\| \geq \epsilon\bigg\},$$ where $U^0$ is the ...
2
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0answers
21 views
Analytic functionals carried by $K$
Let $K$ be a compact subset of $\mathbb{C}$. By definition, one has $$\mathcal{O}'(K) = \left( \varinjlim_{U\supset K} \mathcal{O}(U) \right)',$$ where $U$ are open neighborhoods of $K$. My question ...
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2answers
67 views
A morphism form $G$ to $\mathbb{C}^*$, character what does it represent
I've just begin a course on character theory.
Juste to repeat we say :
Let $G$ be a finite group. Then a character $\chi$ is a morphism from $G \to \mathbb{C}^*$.
We then have some property on ...
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2answers
57 views
Proving that $\phi (T) = T^*$ is an isomorphism between vector spaces
I am tasked with the following:
I am thus tasked with proving:
$1)$ $\phi(T)$ is linear, so that it respects closure under scalar multiplication and addition.
$2)$ $\phi(T)$ is a bijection.
I only ...
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1answer
24 views
Constructing a linear map from annihilator of a subspace to dual of the quotient space
If W is a subspace of V, let V/W denote the quotient of V by W and let (V/W)* denote its linear dual. Construct a non zero linear map from Ann(W) to (V/W)*
(1)From Basis
(2) Canonical
I am weak in ...
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1answer
17 views
How do I find a dual basis for all vectors in $R^3$ such that $v_1-3v_2+2v_3=0?$
How do I find a dual basis for all vectors in $R^3$ such that $v_1-3v_2+2v_3=0?$
I know the "regular" basis $B=\{ (3,1,0), (2,0,-1)\}$. But what is the dual basis?
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1answer
37 views
If we identify $V$ and $U$ with their canonical images in $V^{**}$ and $U^{**}$ prove that the restriction of $T^{**}$ to $V$ coincides with $T$. [duplicate]
Let $T : V \rightarrow U$ be a bounded map between two normed spaces. Let $T^* : U^* \rightarrow V^*$ be defined by $T^*(f) = f\circ T$ for all $f\in U^*$ (the adjoint map).
My Question:
If we ...
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0answers
38 views
Linearity of a map demonstrates that the tensor must have one vector space
Thus, we have shown that $\bar{\bigtriangledown}_a- \bigtriangledown_a$ defines a map of dual vectors at p to tensors of type $(0,2)$ at $p$.
By property (1), this map is linear. Consequently $(\...
1
vote
1answer
43 views
triple dual space and more and more
Let $X$ be a normed space. And Define $X^{(n)}$ by $X^{\overbrace{*****....}^{n\ times}}$ where $X^*$ means the dual space of $X$
My question is :
Is there some space $X$ such that a sequence with ...
2
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1answer
60 views
If $S\subseteq M$ is a submanifold, is there a canonical way to identify $T_{p}^{*}S$ as a subspace of $T_{p}^{*}M$?
I have a few questions. Any thoughts to any one of them will be appreciated.
Suppose $S\subseteq M$ is an embedded submanifold of $M$. There is a convenient characterization of the tangent spaces of $...
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1answer
29 views
Basis for the Dual Space Involving Wedge Products
I'm pretty much stuck on the following problem.
Let $V$ be an $n$-dimensional vector space, and let $\omega\in\Lambda^{2}(V^{*})$. Show that there is a basis $\{e^{1},e^{2},\ldots,e^{n}\}$ of $V^{*...
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2answers
56 views
Eigenvectors in the Dual and Double Dual of T
Let $T$ be a linear operator on a vector space $V$ . Prove that if $x$ is an eigenvector for
$T$, then $\hat{x}$ is an eigenvector for $T
^{tt}$
.
(Where $T^{tt}$ is the double dual of $T$ and $\hat{...
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1answer
93 views
example on dual space and annihilator space
Say we have the set $W = \{(x_1 , \cdots , x_n) \in \mathbb{R}^n | x_1 + 2x_2 + \cdots + nx_n = 0\}$
I want to try finding the dual space and the annihilator space for the subspace $W$ of $\mathbb{R}^...
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0answers
29 views
Existence of $v$ such that $f_i(v) = \delta_{i j} $
Let $f_1,...,f_m \in V^*$ be a linearly independent family ,and F a field. Show that for each $1\leq j\leq m$ there is a $v\in V$ such that $ f_i (v)= \delta_{i j} $ for any $1\leq j\leq m$
I ...
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0answers
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$V=\mathbb{M}_{n}(\mathbb{F})$, $f\in V^{*}$ with $f(AB)=f(BA),$ exists $\lambda\in\mathbb{F}$ with $f(A)=\lambda\textrm{tr}(A)\,\forall \,A\in V$ [duplicate]
Let $V=\mathbb{M}_{n}(\mathbb{F})$ the space of $n\times n$ matrices. Let $f\in V^{*}$ satisfying $f(AB)=f(BA).,$ So, exists unique $\lambda\in\mathbb{F}$ such that $f(A)=\lambda\textrm{tr}(A)\,\...
0
votes
1answer
21 views
$L(V)$ isomorphism $L(V*)$.
Let be $V$ a vector space with finite dimension over F. Show that $\mathscr{L}(V^{*})\cong \mathscr{L}(V)$.
My attemp was try to show that $\mathscr{L}(V^{*})$ is isomorphic with the space $\matrix{M^...
2
votes
0answers
20 views
Predual of $W^*$-subalgebra
I've seen many references claiming that if $\mathcal{N}$ is a $\sigma$-weakly closed *-subalgebra of a von Neumann algebra $\mathcal{M}$, then by taking $\mathcal{N}_\bot:=\{\phi\in\mathcal{M}_*|a(\...
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1answer
9 views
Dual Transformation is 1-1 implies tranformation is onto
I Need to show that $T:V_1\to V_2$ is onto iff $T^*:V_2^*\to V_1^*$ is one one. Here $V_1,V_2$ are vector spaces and $T$ is linear transformation
I do not want to use rank/transpose argument.
Left to ...
3
votes
3answers
78 views
Basis of the Dual Space of Polynomial Spaces
I have the following problem:
Let $V=\mathcal{P}_n(\mathbb{R})$ be the vector space of polynomials of degree $\leq n$. Define $\alpha_k : V\to\mathbb{R}$ by
$$
\alpha_k(p)=\int_{-1}^{1}t^kp(t)dt,\...
1
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0answers
52 views
Why is R* “smaller” than Q* [closed]
Why is the dual space of $\mathbb{R}$ a subset of the dual space of $\mathbb{Q}$?
For the dual space I am looking at every linear functional of the form
$$L:X\rightarrow\mathbb{Q}$$
for $X$ either $\...
0
votes
0answers
24 views
Let, $S\subseteq V^*$ and $\text{Ann}(S)=\{v\in V|f(v)=0\ \forall f\in S\}$
The whole question looks like-
Let, $S\subseteq V^*$ and define $\text{Ann}(S)=\{v\in V|f(v)=0\
\forall f\in S\}$. Prove that,
$\text{Ann}(W_1)+\text{Ann}(W_2)=\text{Ann}(W_1\cap W_2)$ where
$...
1
vote
1answer
30 views
Understanding a definition about double-dual spaces
Let $V$ a $\mathbb{F}$-space, and define the function $\Phi:V\rightarrow V^{**}$ by
$$\Phi(v)(f)=f(v)\quad\; \textrm{for all}\;\quad v\in V,\; f\in V^{*} $$
I didn't understand that definition, ...
0
votes
1answer
24 views
Let $W$ a subspace of $V$. Show that every element of $W^{*}$ is the restriction of an element of $V^{*}$ to $W$.
The question is:
Let $W$ a subspace of $V$. Show that every element of $W^{*}$ is the restriction of an element of $V^{*}$ to $W$.
I think that this question is true only if $\dim(V)<\infty$, ...
1
vote
0answers
41 views
Piecewise function Differentiation in Linear Algebra
Assume $V$ is simply a vector space consisting of real functions $f$ that has $n$th derivative and is 0 outside some bounded real interval. We can make differentiation into $D: V \to V, g \mapsto g'$ ...
1
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1answer
46 views
Derivatives of functions in linear algebra
Let $V = C_{c}^\infty(\mathbb{R})$ a space of smooth functions on $\mathbb{R}$ with compact support. This basically means the space consists of all functions $f: \mathbb{R} \to \mathbb{R}$ such that ...
